# LCVT Latin Centroidal Voronoi Tessellations

LCVT is a C++ library which creates Latin Centroidal Voronoi Tessellation (CVT) datasets.

A Latin Square dataset is typically a two dimensional dataset of N points in the unit square, with the property that, if both the x and y axes are divided up into N equal subintervals, exactly one dataset point has an x or y coordinate in each subinterval. Latin squares can easily be extended to the case of M dimensions, and may be pedantically called Latin Hypersquares or Latin Hypercubes in such a case. Statisticians like Latin Squares, as do experiment designers, and and people who need to approximate scalar functions of many variables.

The fact that the projection of a Latin Square dataset onto any coordinate axis is either exactly evenly spaced, or approximately so (depending on the algorithm), turns out to be an attractive feature for many uses.

However, a CVT dataset in a regular domain, such as the unit hypercube, has the tendency for the projections of the points to cluster together in any coordinate axis. This program is mainly an attempt to explore whether a dataset can be computed using techniques similar to those of a CVT, but with the constraint (whether imposed or expected) that the point projections do not clump up.

The approach used here is quite simple. First we compute a CVT in M dimensions, comprising N points. We assume that the bounding region is the unit hypercube. We are now going to adjust the coordinates of the points to achieve the Latin Hypercube property. For each coordinate direction, we simply sort the points by that coordinate, and then overwrite the original values by the values we'd expect to get for a centered Latin Hypercube, namely, 1/(2*N), 3/(2*N), ..., (2*N-1)/(2*N).

Now this process guarantees that we get a Latin Hypercube. Our hope is that the process of adjusting the point coordinates does not too severely damage the nice dispersion properties inherent in the CVT point placement.

### Languages:

LCVT is available in a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

BOX_BEHNKEN, a C++ library which computes a Box-Behnken design, that is, a set of arguments to sample the behavior of a function of multiple parameters;

CVT, a C++ library which computes a Centroidal Voronoi Tessellation.

FAURE, a C++ library which computes elements of a Faure quasirandom sequence.

GRID, a C++ library which computes elements of a grid dataset.

HALTON, a C++ library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a C++ library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a C++ library which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE, a FORTRAN90 library which computes elements of an angled hexagonal grid dataset.

IHS, a C++ library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a C++ library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a C++ library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a C++ library which computes elements of a Latin Hypercube dataset, choosing points at random.

LATINIZE, a C++ program which "latinizes" a dataset.

LCVT, a dataset directory which contains a collection of sample LCVT datasets.

LCVT_DATASET, a C++ program which creates an LCVT dataset.

NIEDERREITER2, a C++ library which computes elements of a Niederreiter quasirandom sequence with base 2.

NORMAL, a C++ library which computes elements of a normal pseudorandom sequence.

SOBOL, a C++ library which computes elements of a Sobol quasirandom sequence.

UNIFORM, a C++ library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a C++ library which computes elements of a van der Corput quasirandom sequence.

### Reference:

1. Franz Aurenhammer,
Voronoi diagrams - a study of a fundamental geometric data structure,
ACM Computing Surveys,
Volume 23, Number 3, September 1991, pages 345-405.
2. Franz Aurenhammer, Rolf Klein,
Voronoi Diagrams,
in Handbook of Computational Geometry,
edited by J Sack, J Urrutia,
Elsevier, 1999,
LC: QA448.D38H36.
3. John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,
User Manual and Supporting Information for Library of Codes for Centroidal Voronoi Placement and Associated Zeroth, First, and Second Moment Determination,
Sandia National Laboratories Technical Report SAND2002-0099,
February 2002,
../../publications/bgpb_2002.pdf
4. Qiang Du, Vance Faber, Max Gunzburger,
Centroidal Voronoi Tessellations: Applications and Algorithms,
SIAM Review,
Volume 41, Number 4, December 1999, pages 637-676.
5. Michael McKay, William Conover, Richard Beckman,
A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code,
Technometrics,
Volume 21, 1979, pages 239-245.
6. Vicente Romero, John Burkardt, Max Gunzburger, Janet Peterson,
Initial Evaluation of Pure and "Latinized" Centroidal Voronoi Tessellation for Non-Uniform Statistical Sampling,
Sensitivity Analysis of Model Output (SAMO 2004) Conference, Santa Fe, March 8-11, 2004,
rbgp_2004.pdf.
7. Yuki Saka, Max Gunzburger, John Burkardt,
Latinized, improved LHS, and CVT point sets in hypercubes,
submitted to IEEE Transactions on Information Theory,
sgb_submitted.pdf.

### List of Routines:

• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CLUSTER_ENERGY returns the energy of a dataset.
• CVT computes a Centroidal Voronoi Tessellation.
• CVT_ITERATION takes one step of the CVT iteration.
• CVT_WRITE writes a CVT dataset to a file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• FIND_CLOSEST finds the Voronoi cell generator closest to a point X.
• GET_SEED returns a random seed for the random number generator.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_TO_HALTON computes an element of a Halton sequence.
• LCVT_WRITE writes a Latinized CVT dataset to a file.
• PARAM_PRINT prints the program parameters.
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8_EPSILON returns the roundoff unit for R8 arithmetic.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_LATINIZE "Latinizes" an R8MAT.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC.
• REGION_SAMPLER returns a sample point in the physical region.
• S_EQI reports whether two strings are equal, ignoring case.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TEST_REGION determines if a point is within the physical region.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• TUPLE_NEXT_FAST computes the next element of a tuple space, "fast".

You can go up one level to the C++ source codes.

Last revised on 22 September 2006.